Question

solve 4x^2-9x-34=0 using the quadratic formula write answers in decimal forms as needed and give the lesser value first

Answer 1

To solve the quadratic equation \(4x^2 - 9x - 34 = 0\) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\):

- \(a = 4\)
- \(b = -9\)
- \(c = -34\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now, we calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 = (-9)^2 = 81
\]
\[
4ac = 4 \cdot 4 \cdot (-34) = -544
\]
\[
b^2 - 4ac = 81 - (-544) = 81 + 544 = 625
\]

Now we can substitute these values into the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{625}}{2 \cdot 4}
\]

Calculating the square root and simplifying:

\[
x = \frac{9 \pm 25}{8}
\]

Now we will calculate the two possible values for \(x\):

1. Using the positive sign:

\[
x_1 = \frac{9 + 25}{8} = \frac{34}{8} = 4.25
\]

2. Using the negative sign:

\[
x_2 = \frac{9 - 25}{8} = \frac{-16}{8} = -2
\]

Thus, the two solutions for the equation \(4x^2 - 9x - 34 = 0\) are:

\[
x_1 = 4.25 \quad \text{and} \quad x_2 = -2
\]

According to your request to list the lesser value first, the final answer is:

\[
\boxed{-2} \quad \text{and} \quad \boxed{4.25}
\]